Kitaev's model has spins distributed on a honeycomb lattice with nearest-neighbor spin couplings \(J^\alpha \sigma _i^\alpha \sigma _j^\alpha \)  that depend on the bond direction [1]}. In the above, \(\alpha =x,y,z\)  stands for a certain coordinate component in real space, and \(i\)  (\(j\) ) is an index labeling the site in the honeycomb lattice. The model is solved by fermionization; we use Jordan-Wigner strings that are oriented along the \(x,y\)  bonds (see Fig.1). Many Kitaev-inspired models generalize this geometry to include structures such as chains and 2-leg ladders and generalize the interactions to include non-Ising-like couplings such as \(\sigma _i^x \sigma _j^y\) . For an exact solution to exist, the lattice geometry and interactions must conform to a broad set of rules [2]}. We consider two particular models: the original honeycomb lattice and a one-dimensional variant (called the Kitaev spin chain), which is a chain extracted from the original Kitaev honeycomb model by turning off the \(J^y\)  couplings.
